f07chf computes error bounds and refines the solution to a real system of linear equations or , where is an tridiagonal matrix and and are matrices, using the factorization returned by f07cdf and an initial solution returned by f07cef. Iterative refinement is used to reduce the backward error as much as possible.
The routine may be called by the names f07chf, nagf_lapacklin_dgtrfs or its LAPACK name dgtrfs.
3Description
f07chf should normally be preceded by calls to f07cdfandf07cef. f07cdf uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix as
where is a permutation matrix, is unit lower triangular with at most one nonzero subdiagonal element in each column, and is an upper triangular band matrix, with two superdiagonals. f07cef then utilizes the factorization to compute a solution, , to the required equations. Letting denote a column of , f07chf computes a component-wise backward error, , the smallest relative perturbation in each element of and such that is the exact solution of a perturbed system
The routine also estimates a bound for the component-wise forward error in the computed solution defined by , where is the corresponding column of the exact solution, .
4References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
5Arguments
1: – Character(1)Input
On entry: specifies the equations to be solved as follows:
Solve for .
or
Solve for .
Constraint:
, or .
2: – IntegerInput
On entry: , the order of the matrix .
Constraint:
.
3: – IntegerInput
On entry: , the number of right-hand sides, i.e., the number of columns of the matrix .
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array dl
must be at least
.
On entry: must contain the subdiagonal elements of the matrix .
5: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d
must be at least
.
On entry: must contain the diagonal elements of the matrix .
6: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array du
must be at least
.
On entry: must contain the superdiagonal elements of the matrix .
7: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array dlf
must be at least
.
On entry: must contain the multipliers that define the matrix of the factorization of .
8: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array df
must be at least
.
On entry: must contain the diagonal elements of the upper triangular matrix from the factorization of .
9: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array duf
must be at least
.
On entry: must contain the elements of the first superdiagonal of .
10: – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array du2
must be at least
.
On entry: must contain the elements of the second superdiagonal of .
11: – Integer arrayInput
Note: the dimension of the array ipiv
must be at least
.
On entry: must contain the pivot indices that define the permutation matrix . At the th step, row of the matrix was interchanged with row , and must always be either or , indicating that a row interchange was not performed.
12: – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b
must be at least
.
On entry: the matrix of right-hand sides .
13: – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07chf is called.
Constraint:
.
14: – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array x
must be at least
.
On entry: the initial solution matrix .
On exit: the refined solution matrix .
15: – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07chf is called.
Constraint:
.
16: – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the forward error bound for each computed solution vector, such that , where is the th column of the computed solution returned in the array x and is the corresponding column of the exact solution . The estimate is almost always a slight overestimate of the true error.
17: – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the component-wise relative backward error of each computed solution vector (i.e., the smallest relative change in any element of or that makes an exact solution).
18: – Real (Kind=nag_wp) arrayWorkspace
19: – Integer arrayWorkspace
20: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7Accuracy
The computed solution for a single right-hand side, , satisfies an equation of the form
where
and is the machine precision. An approximate error bound for the computed solution is given by
where , the condition number of with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Routine f07cgf can be used to estimate the condition number of .
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f07chf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07chf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The total number of floating-point operations required to solve the equations or is proportional to . At most five steps of iterative refinement are performed, but usually only one or two steps are required.